The renormalization group approach to phase transitions is used to study the mechanism leading to the irrelevance of the quantum nature of a system, as far as its critical behaviour is concerned. The Hamiltonian of an interacting Bose gas is written in the same form as the one appropriate to a 2-component classical system. The quantum nature of the Bose gas is thus completely embodied in the commutation relations. Both in Wilson's and Gell-Mann and Low's formalisms, it is shown that the value of commutation relations is an irrelevant variable and vanishes at the fixed point of the renormalization transformation. The classical Landau-Ginzburg treatment of a 2-component system is thus shown to lead to the same critical exponents as those found for a quantum Bose system, using the renormalization group approach, to every order in perturbation theory.
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